Question: $f(x)=\dfrac{2}{1+2x}$ Find a power series for $f$. Choose 1 answer: Choose 1 answer: (Choice A) A $4-8x+16x^2+\ldots +4(-2)^n x ^n+\ldots$ (Choice B) B $1-2x+4x^2+\ldots +(-2)^n x ^n+\ldots$ (Choice C) C $2+4x+8x^2+\ldots +2(-2)^n x ^n+\ldots$ (Choice D) D $2-4x+8x^2+\ldots +2(-2)^n x ^n+\ldots$
Solution: This is a geometric series with first term $a\text{ }=\text{ }2$ and common ratio $r=-2x\,$. Therefore, the series is as follows. $2-4x+8x^2+\ldots +2(-2)^n x ^n+\ldots $